Hydrodynamic pads cooperate amongst themselves and with oil, or other liquid or gaseous fluid, in the same housing to form an overall bearing for a journal or shaft to be rotated within the housing. The shaft commonly rotates with its axis oriented either vertically or horizontally. Each hydrodynamic pad typically defines a concave arc on its inner face. Further, this arc faces the convex surface of the shaft. Also, a mechanical pivot on the housing often supports the pad.
Instead of the mechanical pivot, one can support the pad using a hydrostatic pivot. An example of such a hydrostatic pivot can be configured in accordance with the disclosure of U.S. Pat. No. 4,059,318 to Hollingsworth, Nov. 22, 1977, which is hereby incorporated herein by reference in its entirety.
A conventional tilting pad is an example of a tilting type of hydrodynamic pad. Furthermore, conventional tilting pads are widely acknowledged to be the most stable type of hydrodynamic pad. An example of an overall bearing configuration for a shaft having horizontal axis orientation could include two lower, tilting pads and one upper, stationary pad, in accordance with the disclosure of U.S. Pat. No. 4,597,676 to Vohr et al., Jul. 01, 1986, which is hereby incorporated herein by reference in its entirety.
Hydrodynamic pads work through a wedge effect in the fluid between the pads and the convex surface of the shaft. Such fluid is often descriptively called the squeeze film. This fluid wedge yields a hydrodynamic lift acting on the convex surface of the shaft and directed away from the arc of the pad. One can recognize occurrences of this wedge effect in common events such as a person water-skiing or an automobile tire hydroplaning.
Introducing preload is a usual method for enhancing the fluid wedge effect. For example, one can assemble the pads to form a circular bearing having a first radius larger than the radius of a particular shaft to be supported. Then, one can remove material from the leading and trailing portions, or edges, of the pads so they physically can be assembled more closely around the convex surface of the shaft. Nevertheless, the arc of each individual pad still corresponds to the first radius. This allows for the preload, as discussed below.
Considering an instance when the shaft is positioned close to an individual pad and symmetrically with respect to the arc of that pad, one can understand that the convex surface of the shaft is physically closer to the arc of the pad at the center of the arc than at either end of the arc. Analysis of the region from the leading edge to the center of the arc reveals this arrangement gives rise to a converging hydrodynamic fluid wedge between the convex surface and the arc.
Introducing offset of the pivot with respect to the center of the arc length of the pad is another means of enhancing the fluid wedge effect. Typically, the pivot can be offset longitudinally downstream between about fifty to sixty percent of the arc length. One can employ offset to increase the fluid wedge effect through modification of the relationship of moments between the leading and trailing lever arms of the pad. Preload, discussed above, can be combined with offset to further increase the fluid wedge effect.
Tilting by the conventional tilting pad further enhances the wedge effect. Namely, the conventional tilting pad desirably accentuates the converging wedge by permitting the leading edge of the pad to pivot away from the convex surface of the shaft.
The overall bearing, formed by the circular arrangement of the pads, has an overall radius. As discussed above with respect to preload, the arc of each pad could correspond to a first radius different from the theoretical overall bearing radius into which the pads are assembled. Eccentricity measures the deviation from an ideal condition in which the axis of the shaft is collinear with the axis of the overall bearing. In this ideal condition, one can say the shaft is centered and experiences zero load. Furthermore, in this ideal condition, the shaft has maximum clearance with respect to the overall bearing.
As one deviates from this ideal condition, into many possible non-ideal conditions, by loading the shaft, eccentricity increases. Moreover, the clearance of the shaft with respect to a particular pad of the bearing decreases. During operation at large loads, the shaft assumes maximal operating eccentricity, assuming a non-failure/non-contact condition. At contact during operation, the eccentricity equals one, so the shaft has zero clearance over the particular pad. This operational failure condition allows the shaft to undergo forced mechanical engaging between its convex surface and the face of the particular pad. Of course, at start-up and shut-down, that is, before and after operational rotation of the shaft, mechanical contact occurs without operational failure.
Hydrodynamic pads are prevalent in turbo-machinery such as pumps, compressors, and turbines. For instance, consider the case of a turbine blade on its shaft. Here, turbine efficiency is determined, in part, by how little clearance one must design for the tip of the rotating blade to pass over the stationary housing. This clearance represents a loss because it provides a fluid leakage path. Namely, fluid passing through the leakage path makes no positive contribution because it escapes work. This loss is characteristic of all turbo-machinery having some type of rotating impeller. The performance or effectiveness of the rotational operation of the turbo-machinery is strongly inversely proportional to the amount of clearance the designer must provide for operation of the impeller or blades mounted on the shaft. So, one desires to minimize the required operating clearance.
During rotation, the shaft tends to orbit elliptically, as is well-known in the art. This elliptical orbit further contributes to the amount of clearance a designer must provide for the operation of the shaft during its rotation. Accordingly, one desires to minimize both the eccentricity of the shaft position and also the ellipticity of its orbit, during rotation.
For dynamic considerations, a convenient representation of bearing characteristics is by spring and damping coefficients. For a horizontal shaft axis orientation, these are obtained as follows.
First, the equilibrium position to support the given load is established by computer solution of the well-known Reynolds equation. Here, horizontal and vertical directions are represented by respective X and Y directions. Second, a small displacement is applied to the shaft in the X direction. A new solution of Reynolds equation is obtained and the resulting forces in the X and Y directions are produced. The spring coefficients are as follows:
where .DELTA.F.sub.x =difference between X forces in the displaced and equilibrium ##EQU1## positions where .DELTA.F.sub.y =difference between Y forces in the displaced and equilibrium positions PA1 where .DELTA.y=displacement from equilibrium position in Y direction PA1 K.sub.xy =stiffness in X direction due to Y displacement PA1 K.sub.yy =stiffness in Y direction due to Y displacement PA1 F.sub.i =force in the i.sup.th direction, where repeated subscripts imply summation, for example: PA1 Shaft Diameter, D=5 in. PA1 Bearing Length, L=5 in. PA1 Active Pad Angle, .theta..sub.p =160.degree. (10.degree. grooves on either side of pad) PA1 Radial Clearance, c=0.0025 in. PA1 Operating Speed, N=5000 rpm PA1 Lubricant Viscosity, .mu.=2.times.10.sup.-6 lb-sec/in..sup.2 PA1 Eccentricity Ratio, .epsilon.=0.5 PA1 Load Direction=Vertical Down PA1 Bearing Load, w=20,780 lbs. PA1 Horsepower Loss, hp=15.51 PA1 Minimum Film Thickness, h.sub.M =0.00125 in. PA1 Side Leakage, q.sub.s =0.941 gpm PA1 Spring and Damping Coefficients: (The signs, positive or negative, of the coefficients conform to the rotor dynamic codes utilized.)
Third, the shaft is returned to its equilibrium position and a Y displacement applied. Next, similar reasoning produces K.sub.xx and K.sub.yx. The damping coefficients D.sub.ij are produced in a like manner. Namely, velocities, rather than displacements, in the X and Y directions are consecutively applied with the shaft in the equilibrium position. So, for most fixed bearing configurations, there are a total of eight coefficients: four spring (or stiffness) and four damping.
The total force on the shaft is: EQU F.sub.i =K.sub.ij X.sub.j +D.sub.ij X.sub.j
The spring and damping coefficients represent a EQU K.sub.ij X.sub.j =K.sub.ix X+K.sub.iy Y
linearization of bearing characteristics. Here, one should determine the equilibrium position accurately because the coefficients are valid only about a small displacement region.
The magnitude of the off-diagonal terms of the spring and damping coefficients matrices reflects the degree of cross-coupling in the bearing configuration. One should note that the matrix of the spring coefficients is commonly referred to as the stiffness matrix. For example, consider the following common geometrical and operating conditions of a single-piece, two axial groove bearing for a horizontal shaft.
For these conditions, a computer solution yields the following results: